To see whether there exist stable bound orbits in the microstate geometries for \(n=3\) , let us focus on the simple case of \(l_{\phi _2}=0\) .
For the three-center solutions, the \(z\) -axis is composed of the 4 intervals,
\(I_-=\lbrace (\rho ,z)|\rho =0,z<z_1\rbrace \) , \(I_1=\lbrace (\rho ,z)|\rho =0,z_1<z<z_{2}\rbrace \) , \(I_2=\lbrace (\rho ,z)|\rho =0,z_2<z<z_{3}\rbrace \)  and \(I_+=\lbrace (\rho ,z)|\rho =0,z>z_3\rbrace \) .
As was previously shown in [1]}, it should be noted that only particles with the angular momentum of \(l_{\phi _2}=0\)  can move on \(I_1\)  and \(I_+\) 
(because \(I_+\)  and \(I_1\)  correspond to the fixed points of the Killing isometry \(v_2:=\partial /\partial \phi _2\) ), and hence only particle with the angular momentum of \(J:=p_\mu v_2^\mu =L_{\phi _2}=0\ (l_{\phi _2}=0)\)  can move on the \(z\) -axis, whereas (because \(U\)  diverges) on \(I_-\)  and \(I_2\)  the particles cannot move.
It can be shown from the entirely similar discussion that on \(I_-\)  and \(I_2\)  only particles with the angular momenta of \(l_{\phi _1}=0\)  are allowed to move there.
